Response of a Floating Ice Plate to a Moving Load

摘要

The steady response of an infinite unbroken floating ice sheet to a moving load is considered. For a concentrated line load, earlier studies based on the linearisation of the problem have shown that there are two 'critical' load speeds near which the steady deflection is unbounded. These two speeds are the speed c_0 of gravity waves on shallow water and the minimum phase speed c_(min). Since obviously deflections cannot become infinite as the load speed approaches a critical speed, Nevel suggested nonlinear effects, dissipation or inhomogeneity of the ice, as possible explanations. The present study is restricted to the effects of nonlinearity when the load speed is close to c_(min). A weakly nonlinear analysis, based on dynamical systems theory and on normal forms, is performed. The difference between the critical speed cm;, and the load speed U is taken as bifurcation parameter. At leading order the problem is reduced to a forced nonlinear Schrodinger equation, which can be integrated exactly. It is shown that the water depth plays a role in the effects of nonlinearity. For large enough water depths, ice deflections in the form of solitary waves exist for all speeds up to (and including) c_(min). For small enough water depths, steady bounded deflections exist only for speeds up to U*, with U* < c_(min). The model is validated by comparison with experimental results in Antarctica (deep water) and in a lake in Japan (relatively shallow water). Finally nonlinear effects are compared with dissipation effects. Our main conclusion is that nonlinear effects play a role in the response of a floating ice plate to a load moving at a speed slightly smaller than c_(min). In deep water, they are a possible explanation for the persistence of bounded ice deflections for load speeds all the way up to c_(min). In shallow water, there seems to be an apparent contradiction, since bounded ice deflections have been observed for speeds up to c_(min) while the theoretical results predict bounded ice deflection only for speeds up to U* < c_(min). But in practice the value of U* is so close to the value of c_(min) that it is difficult to distinguish between these two values.